CHAOS 18, 037102 共2008兲
Network synchronizability analysis: A graph-theoretic approach
Guanrong Chen1,2,a兲 and Zhisheng Duan1,b兲
1
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace
Engineering, College of Engineering, Peking University, Beijing 100871, People’s Republic of China
2
Department of Electronic Engineering, City University of Hong Kong, Hong Kong 220,
People’s Republic of China
共Received 29 February 2008; accepted 9 July 2008; published online 22 September 2008兲
This paper addresses the fundamental problem of complex network synchronizability from a graphtheoretic approach. First, the existing results are briefly reviewed. Then, the relationships between
the network synchronizability and network structural parameters 共e.g., average distance, degree
distribution, and node betweenness centrality兲 are discussed. The effects of the complementary
graph of a given network and some graph operations on the network synchronizability are discussed. A basic theory based on subgraphs and complementary graphs for estimating the network
synchronizability is established. Several examples are given to show that adding new edges to a
network can either increase or decrease the network synchronizability. To that end, some new
results on the estimations of the synchronizability of coalescences are reported. Moreover, a necessary and sufficient condition for a network and its complementary network to have the same
synchronizability is derived. Finally, some examples on Chua circuit networks are presented for
illustration. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2965530兴
First, two simple examples of regular symmetrical graphs
are presented, to show that they have the same structural
parameters (average distance, degree distribution, and
node betweenness centrality) but have very different synchronizabilities. This demonstrates the intrinsic complexity of the network synchronizability problem. Several examples are then provided to show that adding new edges
to a network can either increase or decrease the network
synchronizability. However, it is found that for networks
with disconnected complementary graphs, adding edges
never decreases their synchronizability. On the other
hand, adding one edge to a cycle of size N ⱖ 5 definitely
decreases the network sychronizability. Then, since sometimes the synchronizability can be enhanced by changing
the network structures, the question of whether the networks with more edges are easier to synchronize is addressed. It is shown by examples that the answer is no.
This reveals that generally there are redundant edges in a
network, which not only make no contributions to synchronization but actually may reduce the synchronizability. Moreover, three types of graph operations are discussed, which may change the network synchronizability.
Further, subgraphs and complementary graphs are used
to analyze the network synchronizability. Some sharp
and attainable bounds are provided for the eigenratio of
the network structural matrix, which characterizes the
network synchronizability especially when the network’s
corresponding graph has cycles, bipartite graphs or product graphs as its subgraphs. Finally, by analyzing the effects of a newly added node, some results on estimating
the synchronizability of coalescences are presented. It is
found that a network and its complementary network
a兲
Electronic mail: [email protected].
Electronic mail: [email protected].
b兲
1054-1500/2008/18共3兲/037102/10/$23.00
have the same synchronizability when the sum of the
smallest nonzero and largest eigenvalues of the corresponding graph is equal to the size of the network, i.e.,
the total number of its nodes. Some equilibrium synchronization examples are finally simulated for illustration.
I. INTRODUCTION
Systems composed of dynamical units are ubiquitous in
nature, ranging from physical to technological, and to biological fields. These systems can be naturally described by
networks with nodes representing the dynamical units and
links representing interactions among them. The topology of
such networks has been extensively studied and some common architectures such as small-world and scale-free networks have been discovered.1,2 It has been known that these
topological characteristics have strong influences on the dynamics of the structured systems, such as, epidemic spreading, traffic congestion, collective synchronization, and so on.
From this viewpoint, systematically studying the network
structural effects on their dynamical processes has both theoretical and practical importance.
In the study of collective behaviors of complex networks, the synchronous behavior in particular as a widely
observed phenomenon in networked systems has received a
great deal of attention in the past decades,3–23 e.g., there are
some recently published review articles on network synchronization in the literature.24,25 Oscillator network models have
been commonly used to characterize synchronous behaviors.
In this setting, a synchonizability theorem provided by
Pecora and Carroll3 indicates that the collective synchronous
behavior of a network is completely determined by the network structure. In fact, the network synchronizability is completely determined by two factors: one is the synchronized
18, 037102-1
© 2008 American Institute of Physics
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037102-2
Chaos 18, 037102 共2008兲
G. Chen and Z. Duan
region related to the node dynamics and the inner linking
function; the other is related to the eigenvalues of the network structural matrix. Based on this basic understanding,
the synchronized region problems were studied and disconnected synchronized regions were found in Refs. 4, 22, and
23. On the other hand, the relationships between the network
synchronizability 共the eigenratio of the network topological
matrix兲 and the network structural parameters were studied
in detail in Refs. 6, 11, 14, 16, 20, and 21. Since the synchronizability is correlated with many topological properties,
it is hard to give a direct relationship between the synchronizability and those topological properties. Donetti et al.15
pointed out that a network with optimized synchronizability
should have an extremely homogeneous structure, i.e., the
distributions of topological properties should be very narrow.
Without considering the node and inner-linking dynamics, a network is completely determined by its outer-linking
structure, i.e., the corresponding graph. Algebraic graph
theory has been well studied 共see Refs. 26–34, and references therein兲. In recent years, there are some research
works35–43 which combine the graph theory and complex
networks to study network synchronization; for example,
network synchronizability was analyzed by a graph-theoretic
method in Ref. 35, and the effects of complementary graphs
and graph operations on network synchronizability were
studied in Refs. 37 and 38. It was shown39 that network
synchronizability has no relations with some statistical properties, and a theory of subgraphs and complementary graphs
was established for studying network synchronization in
Refs. 40 and 41. In addition, algebraic graph theory was used
to study consensus problems in vehicle systems.44,45 All
these research works show that better understanding and
careful manipulation of graphs can be very helpful for network synchronization.
Motivated by the above-mentioned works, this paper focuses on the graph-theoretic approach to network synchronization. Clearly, complex networks are closely related to
graphs. Consider a dynamical network consisting of N
coupled identical nodes, with each node being an
n-dimensional dynamical system, described by
N
ẋi = f共xi兲 − c 兺 aijH共x j兲,
i = 1,2, . . . ,N,
共1兲
eigenvalues of A are strictly positive, which are denoted by
0 = ␭1 ⬍ ␭2 ⱕ ␭3 ⱕ ¯ ⱕ ␭N .
共2兲
The dynamical network 共1兲 is said to achieve 共asymptotical兲 synchronization if x1共t兲 → x2共t兲 → ¯ → xN共t兲 → s共t兲, as
t → ⬁. Because of the diffusive coupling configuration, the
synchronous state s共t兲 苸 Rn is a solution of an individual
node, i.e., ṡ共t兲 = f共s共t兲兲. As shown in Ref. 3, the local stability
of the synchronized solution x1共t兲 = x2共t兲 = ¯ = xN共t兲 = s共t兲 can
be determined by analyzing the so-called master stability
equation,
␻˙ = 兵Df关s共t兲兴 + ␣DH关s共t兲兴其␻ ,
共3兲
where ␣ 苸 R, and Df关s共t兲兴 and DH关s共t兲兴 are the Jacobian
matrices of functions f and H at s共t兲, respectively. The largest Lyapunov exponent Lmax of network 共1兲, which can be
calculated from system 共3兲 and is a function of ␣, is referred
to as the master stability function. In addition, the region S of
negative real ␣, where Lmax is also negative is called the
synchronized region. Based on the results of Refs. 3 and 19,
the synchronized solution of dynamical network 共1兲 is locally asymptotically stable if
− c␭k 苸 S,
k = 2,3, . . . ,N.
共4兲
It is well known that if the synchronized region S is
unbounded, in the form of 共− ⬁ , ␣兴, then the eigenvalue ␭2 of
A characterizes the network synchronizability. On the other
hand, if the synchronized region S is bounded, in the form of
关␣1 , ␣2兴, then the eigenratio r共A兲 = ␭2 / ␭N of the network
structural matrix A characterizes the synchronizability. By
condition 共4兲, the larger the 兩␭2兩 or the r共A兲 is, the better the
synchronizability will be, depending on the types of the synchronized region. This paper focuses on the analysis of the
network synchronizability index r共A兲 for the case of bounded
regions from a graph-theoretic approach.
Throughout this paper, for any given undirected graph
G, eigenvalues of G mean eigenvalues of its corresponding
Laplacian matrix. For convenience, notations for graphs and
their corresponding Laplacian matrices are not differentiated,
and networks and their corresponding graphs are not distinguished, unless otherwise indicated.
j=1
where xi = 共xi1 , xi2 , . . . , xin兲 苸 Rn is the state vector of node i,
f共·兲: Rn → Rn is a smooth vector-valued function, constant
c ⬎ 0 represents the coupling strength, H共·兲: Rn → Rn is
called the inner-linking function, and A = 共aij兲N⫻N is called
the outer-coupling matrix or topological matrix, which represents the coupling configuration of the entire network. This
paper only considers the case that the network is diffusively
connected, i.e., the entries of A satisfy
N
aii = −
兺
aij,
i = 1,2, . . . ,N.
j=1,j⫽i
Further, suppose that if there is an edge between node i and
node j, then aij = a ji = −1, i.e., A is a Laplacian matrix. In this
setting, if the graph corresponding to A is connected, then 0
is an eigenvalue of A with multiplicity 1 and all the other
II. RELATIONSHIPS BETWEEN THE NETWORK
SYNCHRONIZABILITY AND NETWORK STRUCTURAL
PARAMETERS
The relationships between network synchronizability index r共A兲 and network structural characteristics such as average distance, node betweenness, degree distribution, clustering coefficient, etc. have been well studied.6,11,14–16 However,
there are also references 36,37,39 showing that for some special
networks the synchronizability has no direct relations with
the network structural parameters, at least some network
characteristics alone cannot determine the synchronizability.
This shows the complexity of the problem.
For a given degree sequence, a construction method for
finding two types of graphs was given in Ref. 36, where one
resultant graph has large ␭2 and r = ␭2 / ␭N, i.e., good synchronizability, and the other has small ␭2 and r = ␭2 / ␭N, i.e., bad
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037102-3
Chaos 18, 037102 共2008兲
Network synchronizability analysis
1
2
6
1
3
5
2
6
4
3
5
4
FIG. 1. Graph G1.
FIG. 3. Graph Gc1.
synchronizability. This shows that the degree sequence by
itself is not sufficient to determine the synchronizability.
Further, two simple graphs G1 and G2 on six nodes,
shown in Figs. 1 and 2, were presented in Ref. 37, where G1
is a typical bipartite graph with many interesting
properties.26,27 These two graphs have the same degree sequence, where the degree of every node is 3; the same average distance 57 ; and the same node betweenness centrality
2,16 but the synchronizability of network G1 is better than
that of network G2 : ␭2共G1兲 = 3 , r共G1兲 = 0.5; ␭2共G2兲
= 2 , r共G2兲 = 0.4.
Remark 1: Although only two six-node graphs are shown
in Figs. 1 and 2, they can be easily generalized to graphs of
size N = 2n with the same conclusion. Suppose that graph G1
is bipartite, which means that it contains two sets of nodes,
each set containing n isolated nodes, and each node in one
set connects to all the nodes in the other set. Graph G2 is
composed of two fully connected subgraphs, each has size n,
where each node in one subgraph to connected by one edge
corresponding node in the other subgraph. In this case, the
smallest nonzero eigenvalue and the largest eigenvalue of G1
are N / 2 and N, respectively, so r共G1兲 = 21 . On the other hand,
the smallest nonzero eigenvalue and the largest eigenvalue of
G2 are 2 and 共N / 2兲 + 2, respectively,27,34 with r共G2兲 = 4 / 共N
+ 4兲 → 0 as N → + ⬁. Therefore, these two graphs have the
same structural parameters but have very different synchronizabilities.
Remark 2: It was pointed out in Ref. 39 that small local
changes in the network structure can sensitively affect the
graph eigenvalues relevant to the synchronizability, while
some basic statistical network properties such as degree distribution, average distance, degree correlation, and clustering
coefficient remain essentially unchanged. Together with the
results discussed above, it is clear that more work is needed
in order to unveil the effects of network structural parameters
on the network synchronizability. According to the existing
results, the statistical properties can distinguish some types
of networks with good or bad synchronizabilities, but there
are also special networks for which the statistical properties
fail to tell any difference between their synchronizabilities.
Therefore, more work is needed in order to truly understand
the essence of the complex network synchronizability.
1
Consider a task of enhancing ␭2 and r by adding some
edges to a graph. For this purpose, the following result is
useful.34
Lemma 1: For any given connected undirected graph G
of size N, its nonzero eigenvalues indexed as in Eq. 共2兲 grow
monotonically with the number of added edges, that is, for
any added edge e, ␭i共G + e兲 ⱖ ␭i共G兲, i = 1 , . . . , N.
Therefore, if only the change of the eigenvalue ␭2 is
concerned, adding edges never decreases the synchronizability. However, for the eigenratio r = ␭2 / ␭N, this is not necessarily true. For example, adding an edge between node 1 and
node 3 in graph G2 共Fig. 2兲, denoted by e兵1 , 3其, leads to a
new graph G2 + e兵1 , 3其, whose eigenvalues are 0, 2.2679, 3,
4, 5 and 5.7321. Thus, r共G2 + e兵1 , 3其兲 = 0.3956 is even smaller
than the original r共G2兲 = 0.4. This means that the synchronizability of network G2 + e兵1 , 3其 is worse than that of network
G2. Adding a new edge between node 1 and node 4
instead, one gets r共G2 + e兵1 , 3其兲 ⬍ r共G2 + e兵1 , 3其 + e兵1 , 4其兲
=0.3970⬍ r共G2兲. This means that, the synchronizability of
network G2 + e兵1 , 3其 + e兵1 , 4其 is better than G2 + e兵1 , 3其, but
1
2
6
3
5
III. EFFECTS OF EDGE ADDITION
AND COMPLEMENTARY GRAPHS
4
FIG. 2. Graph G2.
2
6
3
5
4
FIG. 4. Graph Gc2.
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037102-4
Chaos 18, 037102 共2008兲
G. Chen and Z. Duan
6
1
5
1
2
4
2
6
3
3
5
4
FIG. 5. Graph C6.
FIG. 7. Graph C6o.
still worse than G2. Therefore, by adding edges, the network
synchronizability may increase or decrease. In searching for
a condition under which adding edges may enhance the synchronizability, it was found37 that for networks with disconnected complementary graphs, adding edges never decreases
their synchronizability. For a given graph G, the complement
of G, denoted by Gc, is the graph containing all the nodes of
G and all the edges that are not in G. For example, the
complementary graphs of G1 and G2 in Figs. 1 and 2 are
shown in Figs. 3 and 4, respectively. The following results
are needed for discussing complementary graphs.27,34
Lemma 2: For any given graph G, the following statements hold:
Lemma 3: For any cycle CN with N 共ⱖ4兲 nodes, its
eigenvalues are given by ␮1 , . . . , ␮N 关not necessarily ordered
as in Eq. 共2兲兴 with ␮1 = 0 and
共i兲
共ii兲
共iii兲
共iv兲
␭N共G兲, the largest eigenvalue of G, satisfies
␭N共G兲 ⱕ N.
␭N共G兲 = N if and only if Gc is disconnected.
If Gc is disconnected and has exactly q connected
components, then the multiplicity of ␭N共G兲 = N is
q − 1, 1 ⱕ q ⱕ N.
␭i共Gc兲 + ␭N−i+2共G兲 = N, 2 ⱕ i ⱕ N.
The complementary graph of G1, shown in Fig. 3, is
disconnected. The largest eigenvalue of G1 is 6, which remains the same when the graph receives additional edges.
Hence, combining with Lemmas 1 and 2, one concludes that
the synchronizability of all the networks built on graph G1
never decreases with adding edges, as detailed in Ref. 37.
On the other hand, under what conditions will adding
one edge decrease the synchronizability? It was found41 that
adding one edge to a given cycle with N 共N ⱖ 5兲 nodes definitely decreases the synchronizability. To show this, the following lemma for eigenvalues of cycles is needed.34,36
6
冉 冊
冉 冊
3k␲
N
␮k+1 = 3 −
,
k␲
sin
N
sin
k = 1, . . . ,N − 1.
By the above lemma, one can get the following result for
cycles.41
Theorem 1: For any cycle CN with N ⱖ 4 nodes, adding
one edge will never enhance but possibly decrease its synchronizability r共CN兲; specifically, r共C4 + e兲 = r共C4兲 and
r共CN + e兲 ⬍ r共CN兲.
For example, for N ⱖ 5 adding one edge to cycle C6 with
6 nodes definitely decreases the synchronizability, as shown
in
1
Figs. 5 and 6, where r共C6兲 = 41 = 0.25, r共C6 + e兵1 , 3其兲 = 4.4142
=0.2265⬍ r共C6兲.
However, the synchronizability may be enhanced by
changing the network structure after edge addition. For example, one can change C6 + e兵1 , 3其 to C6o as shown in Fig. 7,
giving r共C6o兲 = 1.2679 / 4.7231= 0.2684, which is better than
r共C6兲 共see Ref. 41 for details兲.
Following above discussions, in optimizing network
structures another interesting question is whether networks
with more edges are easier to synchronize. It was found41
that the answer is negative.
Lemma 4: For any graph G with 16 edges on 10 nodes,
its eigenratio is bounded by r共G兲 ⬍ 52 .
1
1
2
9
2
5
4
3
FIG. 6. Graph C6 + e兵1 , 3其.
3
10
8
4
7
6
5
2
FIG. 8. Graph ⌫1, r共⌫1兲 = 5 .
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037102-5
Network synchronizability analysis
Chaos 18, 037102 共2008兲
FIG. 11. Graph ⌫2.
FIG. 9. Two graphs G = C3 and H = P2.
Lemma 4 shows that there is not a graph G with 16
edges on 10 nodes whose synchronizability is r共G兲 ⱖ 52 .
However, there does exist a graph ⌫1 with 15 edges on 10
nodes whose synchronizability is r共⌫1兲 = 52 共see Fig. 8兲, consistent with the result of Ref. 15. This clearly shows that
networks with more edges are not necessarily easier to synchronize. In fact, by the optimal result of Ref. 15, r = 52 is the
optimal synchronizability for graphs with 15 edges on 10
nodes. For any graph G with 16 edges on 10 nodes, if both G
and Gc have even cycles 共i.e., cycles with even-degree
nodes兲, then by the result of Ref. 40, r共G兲 ⱕ 62 = 31 . Therefore,
adding one more edge definitely decreases the synchronizability in this case.
Remark 3: For simplicity, this section only discusses
some six-node graphs and a ten-node graph. Clearly, for the
edge-addition and structure-changing effects on the network
synchronizability, one can generalize the above discussions
to cycles with N nodes, as shown in Figs. 6 and 7. It is still
an interesting topic for further research to find more examples, as discussed in Lemma 4, to show the effects of
optimizing the network structure by adding or removing
edges.
IV. EFFECTS OF GRAPH OPERATIONS
The effects of graph operations, such as product, join,
coalescence, etc., on network synchronization were studied
in Ref. 38.
First, consider the product operation. Consider two nonempty graphs G共V1 , E1兲 and H共V2 , E2兲. Their Cartesian product graph G ⫻ H is defined, as in Refs. 27 and 38, to be a
graph obtained as follows:
共i兲
共ii兲
the set of nodes of G ⫻ H is the Cartesian product
V1 ⫻ V2; and
any two nodes 共u , u⬘兲 and 共v , v⬘兲 are adjacent in
G ⫻ H if and only if
共ii.1兲 u = v and u⬘ is adjacent with v⬘ in H; or
共ii.2兲 u⬘ = v⬘ and u is adjacent with v in G.
FIG. 10. Generation of a product graph.
Take the two graphs G = C3 and H = P2 in Fig. 9 as an
example. First, establish the Cartesian product space of their
nodes, as shown in Fig. 10共a兲. Then, label the new nodes in
the product space, as shown in Fig. 10共b兲. Finally, connect
some of its node pairs in the product space by following 共ii兲
above, as shown in Fig. 10共c兲, which is isomorphic to graph
G2 shown in Fig. 2, as can be verified by folding the node
共x2 , y 2兲 to the outside of the square, namely, G2 = C3 ⫻ P2.
For two nonempty graphs G and H, it is known that
and
␭max共G ⫻ H兲
␭2共G ⫻ H兲 = min兵␭2共G兲 , ␭2共H兲其
= ␭max共G兲 + ␭max共H兲, so r共G ⫻ H兲 ⬍ min兵r共G兲 , r共H兲其.
Then, consider the join operation. Let G共V1 , E1兲 and
H共V2 , E2兲 be two graphs on disjoint sets of n and m nodes,
respectively. Their disjoint union G + H is the graph G + H
= 共V1 艛 V2 , E1 艛 E2兲, and their joint G * H is the graph on
n + m nodes obtained from G + H by inserting new edges
from each node of G to each node of H, as can be easily
imagined.27,28,38 It is well known that the largest
eigenvalue of the graph G * H is ␭max共G * H兲 = n + m,
and the smallest nonzero eigenvalue is ␭2共G * H兲
= min兵␭2共G兲 + m , ␭2共H兲 + n其 ⱖ ␭2共G兲 + ␭2共H兲.
Therefore,
r共G * H兲 ⱖ 0.5. For example, graph G1 in Fig. 1 can be
viewed as O3 * O3, where O3 is the graph containing three
isolated nodes, which has r共G1兲 = 0.5.
Finally, consider the coalescence operation. A coalescence of two graphs G and H, denoted by G 䊊 H, is a graph
obtained from the disjoint union G + H by identifying a node
of G with a node of H, as can be easily imagined. The coalescence generally does not yield a unique graph.38 It was
shown in Ref. 38 that ␭2共G 䊊 H兲 ⱕ min兵␭2共G兲 , ␭2共H兲其 and
so
r共G 䊊 H兲
␭max共G 䊊 H兲 ⱖ max兵␭max共G兲 , ␭max共H兲其,
ⱕ min兵r共G兲 , r共H兲其. For example, graph ⌫2 in Fig. 11 can be
viewed as the coalescence of chain P4 in Fig. 12 and chain
P2 in Fig. 9, and indeed r共⌫2兲 ⱕ r共P4兲. Obviously, chain P5
can also be obtained as the coalescence of P4 and P2.
V. THEORY OF SUBGRAPHS AND COMPLEMENTARY
GRAPHS FOR ESTIMATING THE NETWORK
SYNCHRONIZABILITY
The theory of subgraphs and complementary graphs are
used to estimate the network synchronizability in Ref. 40.
This section briefly discusses such estimation problems.
For a given graph G共V , E兲, where V and E denote the set
of nodes and the set of edges of G, respectively. A graph G1
is called an induced subgraph of G, if the node set V1 of G1
is a subset of V and the edges of G1 are all edges among
FIG. 12. Chain P4.
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037102-6
nodes V1 in E. The complementary graph of G, denoted by
Gc, is the graph containing all the nodes of G and all the
edges that are not in G.
The following lemma is needed for estimating the largest
Laplacian eigenvalue.26
Lemma 5: For any given connected graph G of size N,
its largest eigenvalue ␭N satisfies ␭N ⱖ dmax + 1, with equality
if and only if dmax = N − 1, where dmax is the maximum degree
of G.
Combining Lemmas 2 and 5, one can obtain the following corollaries.40
Corollary 1: For any given graph G of size N, if its
second smallest eigenvalue equals its smallest node degree,
i.e., ␭2共G兲 = dmin共G兲, then either G or Gc is disconnected; if
␭2共G兲 ⬎ dmin共G兲, then G is a complete graph; if both G and
Gc are connected, then ␭2共G兲 ⬍ dmin共G兲.
Corollary 2: For any given connected graph G and even
its induced subgraph G1, one has ␭max共G兲 ⱖ ␭max共G1兲, so the
synchronizability index of G satisfies
r共G兲 ⱕ
dmin共G兲
;
␭max共G1兲
if both G and Gc are connected, then
r共G兲 ⬍
dmin共G兲
.
dmax共G兲 + 1
Since subgraphs have less nodes, this corollary is useful
when a graph G contains some subgraphs whose largest eigenvalues can be easily obtained.
Corollary 3: For a given graph G, if the largest eigenvalue of Gc is ␭max = dmax共Gc兲 + ␣, then ␭2共G兲
= dmin共G兲 + 1 − ␣. Consequently, the synchronizability index
of G satisfies
r共G兲 =
dmin共G兲 + 1 − ␣ dmin共G兲 + 1 − ␣
ⱕ
.
␭max共G兲
dmax共G兲 + 1
By Lemma 5, generally ␣ ⱖ 1, so the bound in Corollary
3 is better than the one in Corollary 2.
Subgraphs are further discussed below. First, consider
graphs having cycles as subgraphs.
Theorem 2: For any given graph G, suppose H is its
induced subgraph composing of all nodes of G with the
maximum degree dmax共G兲, and Hc is the induced subgraph of
Gc composed of all nodes of Gc with the maximum degree
dmax共Gc兲. Then, if both H and Hc have even cycles 共i.e.,
cycles with even number of nodes兲 as induced subgraphs,
then ␭max共G兲 ⱖ dmax共G兲 + 2 and ␭max共Gc兲 ⱖ dmax共Gc兲 + 2. Consequently, the synchronizability index of G satisfies
r共G兲 ⱕ
Chaos 18, 037102 共2008兲
G. Chen and Z. Duan
dmin共G兲 − 1
.
dmax共G兲 + 2
The smallest even cycle is cycle C4, and its complementary graph Cc4 has two separated edges. C4 and Cc4 are very
important in graph theory.27 A graph has a C4 as an induced
subgraph if and only if Gc has Cc4 as an induced subgraph.
For example, consider graph G2 in Fig. 2. Its complementary
graph is Gc2 in Fig. 4, which is equivalent to C6 in Fig. 5.
1
5
11
2
6
9
12
3
7
10
4
8
FIG. 13. Graph ⌫3.
Testing the eigenvalues of G2 and its synchronizability, one
finds that they attain the exact bounds given in Theorem 2.
For the case of odd cycles as subgraphs, see Ref. 40.
Next, consider graphs having bipartite graphs as subgraphs. A bipartite graph generated by graphs G and H is the
joint G * H of G and H, as discussed in the previous section
共see Refs. 27 and 28 for more details about bipartite graphs兲.
Theorem 3: Let H be a subgraph of a given graph G
containing all nodes of G with the same degree d. Suppose H
contains a bipartite subgraph H1 * H2, and the numbers of
nodes of H1 and H2 are n1 and n2, respectively. Then, the
largest eigenvalue of G satisfies ␭max共G兲 ⱖ d + n1 + n2
− dmax共H1 * H2兲.
For example, consider graph ⌫3 shown in Fig. 13. It can
be easily verified that ⌫3 has a bipartite graph H as its subgraph, which is composed of all nodes with degree 6 from
⌫3. The largest eigenvalue of this bipartite graph is 8. So, by
Theorem 3, ␭max共⌫3兲 ⱖ 9. On the other hand, the maximum
degree of the complementary graph ⌫c3 of ⌫3 is 8. And the
nodes with degree 8 in ⌫c3 form a cycle C4. By Theorem 2,
the smallest nonzero eigenvalue of ⌫3 satisfies ␭2共⌫3兲
ⱕ dmin共⌫3兲 − 1 = 2. So, r共⌫3兲 ⱕ 92 . Simply computing the Laplacian eigenvalues of ⌫3, one obtains ␭2 = 1.7251 and ␭max
= 9.2749. Consequently, r共⌫3兲 = 1.7251 / 9.2749⬇ 0.176.
Therefore, the theorems presented in this section successfully
give the upper integer of the largest eigenvalue and the lower
integer of the smallest nonzero eigenvalue of ⌫3.
Then, consider graphs having product graphs as subgraphs, where the concept of product graphs was introduced
in the previous section.
Theorem 4: For a given graph G, let H be a subgraph of
G containing all nodes of G with the same degree d. Suppose
H contains a product graph H1 ⫻ H2 as its subgraph. Then,
the largest eigenvalue of G satisfies ␭max共G兲 ⱖ d
+ ␭max共H1兲 + ␭max共H2兲 − dmax共H1 ⫻ H2兲.
3
1
15
2
6
13
4
5
9
7
8
11
14
16
10
12
FIG. 14. Graph ⌫4.
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037102-7
Chaos 18, 037102 共2008兲
Network synchronizability analysis
5
1
3
2
4
6
8
7
FIG. 15. Graph ⌫5.
For example, consider graph ⌫4 in Fig. 14. Obviously, all
nodes of ⌫4 have degree 4. And ⌫4 has a product graph C4
⫻ P3 共nodes 1–12兲 as its subgraph, where P3 denotes a chain
with three nodes. The largest eigenvalue of this product subgraph is 7. So, by Theorem 4, ␭max共⌫4兲 ⱖ 7. On the other
hand, the complementary graph ⌫c4 of ⌫4 has a bipartite
graph as its subgraph, which is composed of nodes 1–4 and
nodes 9–12. By Theorem 3, the largest eigenvalue of ⌫c4
satisfies ␭共⌫c4兲 ⱖ dmax共⌫c4兲 + 3. Thus, by Corollary 3, the
smallest nonzero eigenvalue of ⌫4 satisfies ␭2共⌫4兲
ⱕ dmin共⌫4兲 − 2 = 2. So, r共⌫4兲 ⱕ 72 . Simply computing the Laplacian eigenvalues of ⌫4, one obtains ␭2 = 1.2679 and ␭max
= 7.4142. Consequently, r共⌫4兲 = 1.2679 / 7.4142⬇ 0.171.
Similar to the above example, the corresponding theorems
proved in this section successfully give the upper integer of
the largest eigenvalue and the lower integer of the smallest
nonzero eigenvalue of ⌫4.
Finally, consider the maximum disconnected subgraph.
Given a graph G of size N, suppose H is an induced subgraph of G, has size n1, and is disconnected. H is called a
maximum disconnected subgraph, if the node number of any
other disconnected subgraph of G is less than or equal to n1.
Theorem 5: For a given connected graph G of size N, if
the node number of its maximum disconnected subgraph is
n1, then the smallest nonzero eigenvalue of G satisfies ␭2
ⱕ N − n1. Consequently,
r共G兲 ⱕ
N − n1
.
dmax共G兲 + 1
For example, consider graph ⌫5 shown in Fig. 15. By
deleting node 3 or 6, one can verify that the node number of
its maximum disconnected subgraph is 7. So, ␭2共⌫5兲 ⱕ 1.
Combining Lemma 5 and Theorem 5, one obtains r共⌫5兲 ⬍ 51
共see Ref. 40 for details兲.
In fact, for the graphs shown in Fig. 15, one can give a
more precise estimation for their smallest nonzero eigenvalues by using the eigenvector method.27 For example, the
graph 共⌫5 − e兵3 , 6其兲c, i.e., the complementary graph of
⌫5 − e兵3 , 6其, is a bipartite graph, so the eigenvector corresponding
to
its
largest
eigenvalue
is
v
= 共 冑18 , 冑18 , 冑18 , 冑18 , −冑81 , −冑81 , −冑81 , −冑81 兲T.27 Suppose the Laplacian matrix of ⌫5 is L5 共the order of the nodes is as shown in Fig.
15兲. Then, vTL5v = 0.5. By the Releigh-quotient theory of algebraic graph theory,34 one has ␭2共⌫5兲 ⱕ 0.5. By simple computation, one obtains ␭2共⌫5兲 ⬇ 0.3542. Obviously, this new
estimation for ␭2 is sharper than that given by Theorem 5.
It is well known that graphs shown in Fig. 15 have large
node and edge betweenness centralities, therefore have bad
synchronizabilities in general. Based on the theory of subgraphs, complementary graphs and eigenvectors, this section
has given an explanation as why such graphs indeed have
bad synchronizabilities in general.
VI. MORE RESULTS ON THE SYNCHRONIZABILITY
OF COALESCENCES
Given a connected graph G, adding a new node g and a
new edge to connect G and g will form a new graph G + g,
which can be viewed as the coalescence of G and P2 共Fig. 9兲
as discussed in Sec. IV. Now, consider the synchronizability
of G + g. By the result of Ref. 38, r共G + g兲 ⱕ r共G兲. In fact, one
can get more results for this synchronizability estimation, as
shown by the following theorem.
Theorem 6: Given a connected graph G, one has
␭max共G兲 ⱕ ␭max共G 䊊 P2兲 ⬍ ␭max共G兲 + 1 +
1
␭max共G兲
and r共G 䊊 P2兲 ⱕ 1 / ␭max共G兲. If the complementary graph of
G 䊊 P2 is connected, then r共G 䊊 P2兲 ⬍ 1 / ␭max共G兲.
Proof: From the results of Sec. IV, ␭max共G兲
ⱕ ␭max共G 䊊 P2兲 holds obviously. On the other hand, without
loss of generality, suppose that the Laplacian matrix of
G 䊊 P2 is
冢
1
−1
0
冣
L = − 1 d + 1 L12 ,
T
0
L22
L12
where L1 =
共
d
L12
兲
is the Laplacian matrix of G. By the
L22
Schur complement, one has
冋
T
L12
␭max共G兲 + 1 +
册
1
I − L ⬎ 0,
␭max共G兲
if and only if
冢
⌳共G兲 − d −
T
L12
1
⌳共G兲
L12
共⌳共G兲 + 1兲I − L22
冣
⬎ 0,
where ⌳共G兲 = ␭max共G兲 + 1 / ␭max共G兲. Since ␭max共G兲 is the
largest eigenvalue of L1, the above inequality holds obviously. Hence, ␭max共G 䊊 P2兲 ⬍ ␭max共G兲 + 1 + 1 / ␭max共G兲. Further, by Corollary 2 in Sec. V, the last part of the theorem can
be verified.
䊏
In addition, in Theorems 5 and 6, one can give an estimation of the smallest nonzero eigenvalue of G 䊊 H.
Theorem 7: Given two connected graphs G and H, one
has ␭2共G 䊊 H兲 ⱕ 1.
Proof: Let the node numbers of G and H be n and m,
respectively, and g be the identifying node of G in the coalescence of G and H. By deleting node g, the maximum
disconnected subgraph H1 of G 䊊 P2 will have n + m − 1
䊏
nodes. Here, by Theorem 6, ␭2共G 䊊 H兲 ⱕ 1.
By Theorems 6 and 7, one knows that generally the synchronizability of the coalescences of graphs is weak. Let KN
denote a complete graph of size N. Consider the coalescence
of KN and P2 共shown in Fig. 9兲. Obviously, the complemen-
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037102-8
Chaos 18, 037102 共2008兲
G. Chen and Z. Duan
FIG. 18. Graph ⌫8.
FIG. 16. Graph ⌫6.
tary graph of KN 䊊 P2 is disconnected, so ␭max共KN 䊊 P2兲
= N + 1. Further, by Lemma 2, one knows that ␭2共KN 䊊 P2兲
= 1, so the synchronizability index is r共KN 䊊 P2兲 = 1 / N + 1.
But, on the other hand, r共KN兲 = 1. Therefore, in this case, a
newly added node severely decreases the synchronizability.
For example, r共⌫6兲 = 61 , where ⌫6 is shown in Fig. 16. In
addition, graph ⌫5 in Fig. 15 can be viewed as a coalescence
K4 䊊 P2 䊊 K4, which also has weak synchronizability. In order to improve the synchronizability of such graphs, their
graph structures must be modified.24
In what follows, consider the coalescences of starshaped graphs and P2 共shown in Fig. 9兲. Let SN denote a
star-shaped graph of size N. For S6, shown in Fig. 17, it can
be viewed as a coalescence of S5 and P2 by identifying the
central node of S5 and one node of P2. In this case, r共S6兲
= 61 ⬍ r共S5兲 = 51 . Compared with graph KN 䊊 P2, one added
node does not result in severe decrease of the synchronizability of S6. As another example, ⌫8 in Fig. 18 can also be
viewed as a coalescence of S5 and P2. By Lemma 5 and
Corollary 1, ␭max共⌫8兲 ⬎ 5 and ␭2共⌫8兲 ⬍ 1. On the other hand,
⌫8 can also be viewed as a coalescence of P4 and S3, so
␭2共⌫8兲 ⱕ ␭2共P4兲 = 0.5858. Therefore, r共⌫8兲 ⬍ 0.5858 / 5. By
simple computation, one obtains r共⌫8兲 = 0.4859 / 5.0861,
which is worse than r共S6兲. From these examples, one can see
that the effects of adding one node and one edge 共i.e., coalescing P2 to a given graph兲 on the synchronizability are very
different for different graphs. How to reconstruct a new
graph by adding a new node to surely improve the synchronizability is still an interesting open problem.
VII. CONDITIONS FOR A NETWORK AND ITS
COMPLEMENTARY NETWORK TO HAVE THE SAME
SYNCHRONIZABILITY
From the above discussions, one can see that complementary graph is very important for the study of the network
synchronization. Therefore, it is interesting to consider the
synchronizabilities of a network and its complementary network simultaneously.
Theorem 8: For a given connected graph G of size N, if
both G and Gc are connected, then G and Gc have the same
synchronizability if and only if ␭2共G兲 + ␭N共G兲 = N, i.e.,
␭2共G兲 = ␭2共Gc兲 and ␭N共G兲 = ␭N共Gc兲.
Proof: In Lemma 2, G and Gc have the same synchronizability, i.e.,
␭2共G兲 N − ␭N共G兲
=
.
␭N共G兲 N − ␭2共G兲
So, N␭2共G兲 − ␭22共G兲 = N␭N共G兲 − ␭N2 共G兲, i.e., N关␭N共G兲 − ␭2共G兲兴
=关␭N共G兲 + ␭2共G兲兴关␭N共G兲 − ␭2共G兲兴. Since Gc is connected,
␭N共G兲 − ␭2共G兲 ⫽ 0. Therefore, ␭N共G兲 + ␭2共G兲 = N. Thus,
Lemma 2 leads to the conclusion.
It is well known that the complementary graphs of chain
P4 共shown in Fig. 12兲 and cycle C5 are the same as P4 and
C5, respectively. Hence, P4 and C5 satisfy the condition
given in Theorem 8. In addition, graph C6o shown in Fig. 7
satisfies ␭2共G兲 + ␭6共G兲 = 6, so C6o and its complementary
c
have the same synchronizability. Further, delete
graph C6o
the edge e兵3 , 6其 from graph G2 in Fig. 2, and denote the
resultant graph by G2 − e兵3 , 6其. Then, by simple computation,
one can verify that G2 − e兵3 , 6其 and its complementary graph
have the same synchronizability, where r共G2 − e兵3 , 6其兲 = 0.2.
From these examples, one knows that there do exist many
graphs which have the same synchronizability as their
complementary graphs.
Next, consider the effects of edge-adding on the synchronizabilities of a graph and its complementary graph.
First, consider graph C6o + e兵3 , 5其, where C6o is given as in
Fig. 7. By simple computation, one obtains that
r共C6o + e兵3,5其兲 =
1.2679
1.2679
⬍ r共C6o兲 =
,
5.4142
4.7321
and obviously
r共共C6o + e兵3,5其兲c兲 =
FIG. 17. Graph ⌫7 = S6.
0.5858
1.2679
c
兲=
⬍ r共C6o
.
4.7321
4.7321
This means that adding an edge to C6o decreases the sync
chronizabilities of C6o and C6o
simultaneously. On the other
hand, by adding the edge e兵3 , 6其 to G2 − e兵3 , 6其, one can find
that this added edge increases the synchronizabilities of G2
− e兵3 , 6其 and 共G2 − e兵3 , 6其兲c simultaneously. Of course, there
are other examples in which adding one edge increases the
synchronizability of either the original graph or the complementary graph. This shows the complexity in this kind of
edge-adding problems.
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037102-9
Chaos 18, 037102 共2008兲
Network synchronizability analysis
FIG. 20. Network on C6 + e兵1 , 3其.
FIG. 19. Network on G1.
It should be pointed out that although Theorem 8 gives a
condition for a graph and its complementary graph to have
the same synchronizability, the eigenvalue conditions therein
are generally not easy to test. It is more interesting to give a
new condition for this problem which is solely based on
graph characteristics. This leaves an open problem for future
research.
VIII. EQUILIBRIUM SYNCHRONIZATION OF A CHUA
CIRCUIT NETWORK
Consider network 共1兲 consisting of the third-order
smooth Chua’s circuits,46 in which each node is described by
3
+ bxi1兲,
ẋi1 = − k␣xi1 + k␣xi2 − k␣共axi1
共5兲
ẋi2 = kxi1 − kxi2 + kxi3 ,
ẋi3 = − k␤xi2 − k␥xi3 .
Linearizing Eq. 共5兲 about its zero equilibrium gives
ẋi = Fxi,
F=
冢
− k␣ − k␣b
k
0
k␣
0
−k
k
− k␤ − k␥
冣
,
共6兲
where xi = 共xi1 , xi2 , xi3兲T.
Take k = 1 , ␣ = −0.1, ␤ = −1 , ␥ = 1 , a = 1 , b = −25. Then, F
is stable, i.e., the node system 共5兲 is locally stable about zero.
Further, take the inner linking matrix
H=
冢
0.8348
9.6619
2.6591
0.1002
0.0694
0.1005
− 0.3254 − 7.0837 − 0.8042
冣
.
Then, by simple computation, one can verify that F + ␣H has
two disconnected stable regions: S1 = 关−0.01, 0兴 and S2
= 关−3.3, −0.74兴, so the entire synchronized region is S
= S1 艛 S2. Moreover, let N = 6 and the outer coupling matrix A
be equal to the Laplacian matrix G1 shown in Fig. 1. According to the eigenvalues of G1 given in Sec. II, one may take
1
the coupling strength c = 2.1
. Then, for all eigenvalues of G1,
one has c␭i 苸 S2, i = 1 , . . . , 6. By condition 共4兲, network 共1兲
specified with the above data achieves synchronization. Figure 19 shows the state of node 1 in this network. The other
nodes behave similarly.
With the above node dynamics and inner linking function, similar synchronization behaviors can be observed on
other graphs with suitable coupling strengths c, for example,
1
1
1
兲, Gc2 共c = 1.3
兲, C6o 共c = 1.5
兲, and C6 + e兵1 , 3其 共c
on G2 共c = 2.1
1
= 1.345 兲. Figure 20 shows a similar synchronization result on
graph C6 + e兵1 , 3其 共shown in Fig. 6兲. Obviously, the network
on G1 synchronizes much faster than that on C6 + e兵1 , 3其.
This also demonstrates that the synchronizability of G1 is
better than that of C6 + e兵1 , 3其. In fact, the region of the coupling strength c is very narrow for achieving synchronization
of C6 + e兵1 , 3其.
However, for the outer coupling matrix G2 − e兵3 , 6其 or
共G2 − e兵3 , 6其兲c, for any coupling strength c 苸 关0.003, + ⬁ 兲,
condition 共4兲 does not hold. Therefore, for the above case of
node equation, inner coupling matrix and coupling strength,
the network built on G2 − e兵3 , 6其 does not synchronize at all.
Figure 21 shows the state of node 1 in this network with c
1
= 1.8
; the other nodes behave similarly.
For simplicity, this section only discusses some synchronization and nonsynchronization behaviors of Chua’s circuit
networks on six-node graphs. However, similar synchronization problems can be discussed on some graphs with N
nodes. For example, with the node equation as given in Eq.
共5兲 and with the above data, for any natural number n, a
network of N = 2n nodes in type G1 共Fig. 1兲, as discussed in
Remark 1, achieves synchronization with the coupling
strength c = 1 / n. However, a network of N = 2n 共n ⱖ 5兲 nodes
in type G2 共Fig. 2兲, as discussed in Remark 1, cannot achieve
synchronization with any coupling strength c ⬎ 0.01 / n. This
also verifies the statement given in Remark 1 that the synchronizability index of networks of type G2 共Fig. 2兲 tends to
0 as n → + ⬁.
FIG. 21. Network on G2 − e兵3 , 6其.
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037102-10
IX. CONCLUSION
From both geometric and algebraic points of view, the
study on the synchronizability of complex networks can be
separated into two parts: one is on the geometric synchronized region,3,4,22,23,37 for which the larger the synchronized
region the better the synchronizability; the other is the algebraic eigenratio r = ␭2 / ␭N of the corresponding Laplacian
matrix, for which the larger the r the better the
synchronizability.3,7,11,15,16,21,24 This paper has taken both
viewpoints from the graph-theoretic approach to discuss the
performance of the network synchronizability, showing an
in-depth application of the graph theory to network synchronization studies. Sections II–V introduce the existing results
to show the effects of network statistical properties, subgraphs, complementary graphs, graph operations, adding
edges, and adding nodes, etc.15,36–41 In Secs. VI and VII,
some new results on the synchronizability of coalescence
have been introduced, and a condition for a network and its
complementary network to have the same synchronizability
has been illustrated. In Sec. VIII, a Chua’s circuit network
was used to show its synchronization and nonsynchronization behaviors on different graphs and complementary
graphs. The study of this paper has demonstrated that better
understanding and careful manipulation of the underlying
graphs are indeed very important and helpful for investigating complex network synchronization.
ACKNOWLEDGMENTS
This work is jointly supported by the National Natural
Science Foundation of China under Grant No. 60674093, the
Key Projects of Educational Ministry under Grant No.
107110, and the NSFC-HK Joint Research Scheme under
Grant No. N-CityU 107/07.
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